CAMP Special Lecture
This series of lectures is free and open to everyone. If you would join this, in order to let us know, please register here before June 14. Title. Alternating Permutations
 Lecturer. Richard P. Stanley, MIT, USA
 Date. June 29 (Mon.)  July 3 (Fri.), 2015
 Time. 14:00 (June 2930, July 1, July 3), 15:30 (July 2)
 Venue. CAMP & NIMS
 Detailed Time & Venue.
 NIMS Seminar Room (3rd floor), 2:00 PM, June 29, 2015
 NIMS Seminar Room (3rd floor), 2:00 PM, June 30, 2015
 CAMP Seminar Room (M), 2:00 PM, July 1, 2015
 CAMP Seminar Room (L), 3:30 PM, July 2, 2015
 CAMP Seminar Room (L), 2:00 PM, July 3, 2015
 Abstract. (for all lectures) A permutation a_{1} a_{2} ... a_{n} of 1, 2, ..., n is alternating if a_{1} > a_{2} < a_{3} > a_{4} < ... . Alternating permutations have many fascinating properties related to combinatorics, geometry, representation theory, probability theory, and other areas. We will survey the highlights of this subject.
Lecture 1. (June 29, 14:00) Euler numbers and André's theorem
The number of alternating permutations of 1, 2, ..., n is denoted E_{n} and is called an Euler number. The first theorem on alternating permutations is the generating function
∑_{n}_{≥}_{0} E_{n} x^{n} / n! = tan(x) + sec(x) due to D. André in 1879. Some other combinatorial objects are also counted by the Euler numbers.
Lecture 2. (June 30, 14:00) Convex polytopes and alternating permutations
There are two interesting convex polytopes whose volume is E_{n} / n!. One of them is given by x_{i} ≥ 0 (1 ≤ i ≤ n) and x_{i} + x_{i+1} ≤ 1 (1 ≤ i ≤ n1). They fit into a more general context of order polytopes and chain polytopes of posets.
Lecture 3. (July 1, 14:00) Longest alternating subsequences
What can one say about the length of the longest alternating subsequence of a random permutation? This theory is analogous but simpler than the betterknown theory of the longest increasing subsequence of a random permutation. For example, if n>1 then the expected length of the longest alternating subsequence of a random permutation of 1, 2, ..., n is exactly (4n+1)/6.
Lecture 4. (July 2, 15:30) A symmetric group character related to alternating permutations
H. O. Foulkes defined a certain character χ of the symmetric group S_{n} whose values χ(w) are either 0 or ±E_{k} for some 1 ≤ k ≤ n (depending on w). This character allows us to enumerate certain classes of alternating permutations using representation theory. For instance, if p is prime then the number of alternating permutations in S_{p} that are also pcycles is equal to
(E_{p}  (1)^{(p1)/2}) / p. No elementary proof is known.
Lecture 5. (July 3, 14:00) The cdindex of the symmetric group
The cdindex of S_{n} is a noncommutative polynomial in the indeterminates c and d that encodes a lot of combinatorial information. The number of terms in this polynomial is E_{n}, and there are interesting connections with alternating permutations.
ReferencesParticipants 
Name  Affiliation  Position 

Byunghak Hwang  Seoul National University  Student  Heesung Shin  Inha University  Professor  James Haglund  University of Pennsylvania, USA  Professor  Jiang Zeng  Université Claude Bernard Lyon 1, France  Professor  Jinha Kim  Seoul National University  Student  Jung Seok Oh  Seoul National University  Student  KangJu Lee  Seoul National University  Student  Kyoungsuk Park  Ajou University  Student  Masao Ishikawa  University of the Ryukyus, Japan  Professor  Minki Kim  KAIST  Student  Richard P. Stanley  MIT, USA  Professor  SangHoon Yu  Seoul National University  Student  Sangwook Kim  Chonnam National University  Professor  Sanha Lee  Sungkyunkwan University  Student  SeungIl Choi  Sogang University  Postdoc  Seung Jin Lee  KIAS  Postdoc  Shishuo Fu  Chongqing University, China  Professor  Sunyoung Nam  Sogang University  Student  YoungHun Kim  Sogang University  Student  Younjin Kim  KAIST  Postdoc  Zhicong Lin  NIMS  Postdoc 
Showing 21 items
